Thomas Wolff Memorial Lecture in Mathematics
Peter Jones, James E. English Professor of Mathematics and Applied Math at Yale, is a specialist in the field of complex and harmonic analysis, probability theory, and dynamical systems. He came to Yale in 1985 after teaching for six years at the University of Chicago. For several years he lived in Sweden, where he served as assistant director of the Institut Mittag-Leffler. He was the Goran Gustafsson Professor at the Royal Institute of Technology; KTH, in Sweden in 1990. At Yale, he was director of graduate studies in mathematics 1993-95. In 1994, Jones became the youngest person to receive an honorary degree from KTH for his "pathbreaking scientific contributions to modern mathematic analysis" and for promoting the study of mathematics at the institute. He continues to maintain strong ties with the Swedish mathematical community. Jones' other honors include a Sloan Foundation Fellowship, the Salem Prize and a Presidential Young Investigator Award.
Abstract: Rademacher's Theorem states that a Lipschitz mapping from one Euclidean space to another is differentiable except on a set of (Lebesgue) measure zero. In this lecture and the next I will discuss joint work with Marianna Csornyei: Given a set E of measure zero, there is a map from Euclidean space to itself that is not differentiable on E. This is an old and easy result in dimension D = 1. In two dimensions this result was proven by G. Alberti, M. Csornyei, and D. Price. Their proof has two parts. First there is a combinatorial argument (special to dimension = 2), and a "covering argument", which works in any dimension. In this will explain some of the technical tools that go into the work of Csornyei and PJ. Critical here is the notion of the concept of "Tangent Cones" for sets of measure = 0. I will also review some basic machinery from harmonic analysis that is used in our proof.